Rate of heat loss? Speed of heat.

[Gerenuk]

I have some extended aluminum part which is at low temperatures. It is connected by a thin alu rod to a cooling device held at a constant temperature.
How can I estimate the rate of cooling through this thin rod given the material specific constants?

I assume the heat exchange with the surrounding atmosphere is small, but wouldn't mind including it in my calculations.

The Green function is a particle diffusion function. So can I associate the particle/heat speed somehow with a velocity that would give me the "speed of heat"? Then for 1D or 3D I'd calculate the "distance traveled by heat" keeping in mind some diffusion correction depending on the dimension?!

 

[John Creighto]

I have some extended aluminum part which is at low temperatures. It is connected by a thin alu rod to a cooling device held at a constant temperature.
How can I estimate the rate of cooling through this thin rod given the material specific constants?

I assume the heat exchange with the surrounding atmosphere is small, but wouldn't mind including it in my calculations.

The Green function is a particle diffusion function. So can I associate the particle/heat speed somehow with a velocity that would give me the "speed of heat"? Then for 1D or 3D I'd calculate the "distance traveled by heat" keeping in mind some diffusion correction depending on the dimension?!

It would seem to me the most important parameters are the specific heat of the aluminum part, and the thermal conductivity of the metal rod. It sounds like you are trying to kill a fly with a sledgehammer.

 

[gel]

You can't associate any speed to diffusions such as Brownian motion, because of the way they scale under rescaling time.

 

[John Creighto]

You can't associate any speed to diffusions such as Brownian motion, because of the way they scale under rescaling time.

I'm not sure what you mean.

 

[John Creighto]

If we're talking about solids then the particles only vibrate. It is the phonons in the solid which transmit heat. I presume you could do something like radiative transfer but instead of light in a gas consider phonons in a solid, where the scattering would obay something like Beer's law.

 

[gel]

I'm not sure what you mean.

I mean that if X(t) is a Brownian motion then $a^{-1}X(a^2t)$ is also a BM. However, if there was some speed you could naturally associate with X, it should multiply by $a$ under the same transformation.

Equivalently, if f(t,x) is a solution of the heat equation, then so is $f(a^2t,ax)$, but velocities/speeds are not invariant under rescaling space & time like this, because velocity has units of length over time.

Edit:
that is, there is no fixed speed.
I suppose you could define a speed v such that $df/dt+v\cdot\nabla f=0$, but it would depend on f.

 

[Cyrus]

What does brownian motion have to do with this problem? This is basic heat transfer via conduction and convection. Just set up a thermal resistance network and solve. Jeesh.

 

[gel]

What does brownian motion have to do with this problem? This is basic heat transfer via conduction and convection. Just set up a thermal resistance network and solve. Jeesh.

because probabilities for BM satisfy the heat equation. As the OP mentioned particle diffusions, I thought he must be referring to this.

 

[Cyrus]

I think he's trying to use that because it mentions particle speed, to somehow relate it to heat speed, when all he has to do is look at the heat transfer rate.

I could be wrong though. I read it as a solid aluminum bar being cooled on one end. No particle motions are necessary.

 

[John Creighto]

because probabilities for BM satisfy the heat equation. As the OP mentioned particle diffusions, I thought he must be referring to this.

I'm not very good at vector aspects of physics. I did however, read though that the laplacian somehow represents diffusion. I'm not sure why but doesn't the wave equation also depend on the laplacian? I know nothing about how Brownian motion relates to the heat equation but I thought the poster might somehow be interested in transitive properties of heat transfer.

 

[John Creighto]

Isn't Brownian motion just a random walk? Was is that relevant to heat transfer given that energy is exchanged between particles?

 

[gel]

I'm not very good at vector aspects of physics. I did however, read though that the laplacian somehow represents diffusion. I'm not sure why but doesn't the wave equation also depend on the laplacian?

Yes, but the wave equation also has second order derivatives in time, unlike the heat equation which only has a first order time derivative, which makes the solutions very different.
If you have $d^2f/dt^2=k\nabla^2 f$, as in the wave equation, then k has units of velocity^2. In the heat equation you have $df/dt=k\nabla^2 f$ and k has units of length^2/time.

 

[gel]

Isn't Brownian motion just a random walk? Was is that relevant to heat transfer given that energy is exchanged between particles?

Well you calculate heat flow with a monte carlo simulation. Don't know if anyone does this in physics, but you could. If you start a BM off randomly distributed according to the temperature at the start, then its probability density evolves in the same say as temperature does.

 

[Integral]

To the first order this is a simple problem in conduction, You need the thermal properties, cross sectional areas and the lengths of your various components. If you set up the more complex methods mentioned, which could be done, your solution will be very nearly the conduction equation.

Unless there is a large temperature difference to the atmosphere or you are blowing air across the surfaces, convective and radiative losses will be small to negligible.

 

[Gerenuk]

...
Equivalently, if f(t,x) is a solution of the heat equation, then so is $f(a^2t,ax)$, but velocities/speeds are not invariant under rescaling space & time like this, because velocity has units of length over time.
...

Good that at least someone understands the question. I can see that the Green function has a $r^2/t$ dependence, so it's diffusion. Then I wondered: "Do Brownian particles have a velocity associated with them"? If so, it should be possible to find a velocity for heat by analogy. Of course this velocity would have to be corrected to include the diffusion character.

 

[Gerenuk]

To the first order this is a simple problem in conduction, You need the thermal properties, cross sectional areas and the lengths of your various components. If you set up the more complex methods mentioned, which could be done, your solution will be very nearly the conduction equation.

The difficulty is that that shape of the aluminium block on the rod is complicated or maybe to first approximation a plate.

I tried to solve the problem and assumed that the heat spreads instantly in this alu block and that along the rod there is always a constant gradient. I get an exponential solution for the temperature of the alu block.
$$T=A+Be^{-kt}$$

Are these approximations OK?

 

[GT1]

I have some extended aluminum part which is at low temperatures. It is connected by a thin alu rod to a cooling device held at a constant temperature.
How can I estimate the rate of cooling through this thin rod given the material specific constants?

I assume the heat exchange with the surrounding atmosphere is small, but wouldn't mind including it in my calculations.

You don't need to work hard - what you describe is a fin with a constant base temperature. the fin conducts heat and loses (or gains) heat to the environment by convection.
You can find the solution of the differential equation for this problem in any heat transfer book. all you need to do is to put your numbers and you have the answer.

 

[Gerenuk]

You don't need to work hard - what you describe is a fin with a constant base temperature. the fin conducts heat and loses (or gains) heat to the environment by convection.
You can find the solution of the differential equation for this problem in any heat transfer book. all you need to do is to put your numbers and you have the answer.

Hmm, most of the time answer along the lines of "It's very easy. You can find it in any book!" are not helpful. At least some specific reference would be helpful.

Anyway, what you describe isn't the actual difficulty that I saw in the problem. The question is how to treat the boundary conditions where the alu block is. And if some approximation can spare me a detailed numerical simulation. I also don't have a fin, but a rod. The alu block might be a plate, but heat exchange with the environment or convection isn't important either. It's about the heat exchange of the solid parts connected to the cooling device.

 

[GT1]

Hmm, most of the time answer along the lines of "It's very easy. You can find it in any book!" are not helpful. At least some specific reference would be helpful.

Anyway, what you describe isn't the actual difficulty that I saw in the problem. The question is how to treat the boundary conditions where the alu block is. And if some approximation can spare me a detailed numerical simulation. I also don't have a fin, but a rod. The alu block might be a plate, but heat exchange with the environment or convection isn't important either. It's about the heat exchange of the solid parts connected to the cooling device.

Try:
1. Holman, J.P. Heat Transfer
2. Introduction to heat transfer/ Frank P. Incropera, David P. DeWitt.
They are both recommended.
and if you want to ignore the convection to the environment (i don't know if you should) -the cooling power of your device Q [W] - is the amount of energy that is removed from the plate,and then use the equation M*Cp*DT/Dt=Q to find t (cooling time).

 

[russ_watters]

I'm still not seeing any complexity at all in this problem either. The fact that the part is nonuniform shape just means you need to do a finite element analysis, but since the equation is linear, you simply plug-and-chug in an Excel spreadsheet. The toughest part could be entering the cross sectional area profile if it is a very complex shape. Here's the equation and a sample problem:

http://www.engineeringtoolbox.com/conductive-heat-transfer-d_428.html

 

[Gerenuk]

Try:
1. Holman, J.P. Heat Transfer
2. Introduction to heat transfer/ Frank P. Incropera, David P. DeWitt.
They are both recommended.
and if you want to ignore the convection to the environment (i don't know if you should) -the cooling power of your device Q [W] - is the amount of energy that is removed from the plate,and then use the equation M*Cp*DT/Dt=Q to find t (cooling time).

Thanks for the references. I haven't found these books yet, but I'll make a note. Hope there is a similar problem solved.

Your cooling equation wouldn't work in my case, because the whole point is that the temperature is non uniform. The cooling power isn't constant either, since only the temperature of the cooling device is given.

 

[Gerenuk]

I'm still not seeing any complexity at all in this problem either. The fact that the part is nonuniform shape just means you need to do a finite element analysis, but since the equation is linear, you simply plug-and-chug in an Excel spreadsheet. The toughest part could be entering the cross sectional area profile if it is a very complex shape. Here's the equation and a sample problem

I think you didn't understand the question. Have a look at
http://en.wikipedia.org/wiki/Heat_equation#List_of_Green_function_solutions_in_1D
for the type of problem that is here.

The shape of the block isn't specified exactly. And of course there is the possibility to simulate the differential heat equation for an arbitrary shape, but that's not feasible for this problem. Results more approximate are acceptable.

 

[russ_watters]

That describes the establishement of the temperature gradient. It sounds to me like you want the steady state heat transfer rate.

 

[Gerenuk]

That describes the establishement of the temperature gradient. It sounds to me like you want the steady state heat transfer rate.

No, the temperature at one end of the rod is constant. The heat transfer rate will depend on the time-dependent temperature in the rod. I'm interested in how long it takes to reach the steady state, where all temperatures are equal.

 

[John Creighto]

Is the heat equation only true if phonon scattering is large? Are there any macroscopic materials for which the average mean path of the phonon's greater then say 1cm? As I understand phonon's scatter when there is material impurities. How pure are the purist bulk lattice structures?

 

[russ_watters]

No, the temperature at one end of the rod is constant. The heat transfer rate will depend on the time-dependent temperature in the rod. I'm interested in how long it takes to reach the steady state, where all temperatures are equal.

Generally, problems like this have the temperature at both ends of the rod constant (or Q constant and you find the resulting temperaures). It helps a lot to clearly define what you are trying to do here. Not knowing the pracitcal purpose of this exercise makes it difficult to see why this approach is relevant. The transient problem is quite rare in the thermodynamics of heat sinks - almost everying you see, from a car radiator to a computer CPU involves a steady state heat dissipation.

All that aside...
The transient heat loss problem is more complicated and is a differential equation, but I'm still not sure off the top of my head why you can't still do a finite element analysis in Excel (it is 2:00 and I have a few beers in me...). Heat transfer is proportional to delta-T and you can easily enough enter that into a spreadsheet. It's just adding another term - another dimension - to the spreadsheet to reference back to the previous delta-T. The steady-state is essentially one-dimensional, but the transient is two-dimensional.

For steady-state, you can easily do length along your sample vertically. For the transient case, you would have the length elements horizontally, with time vertically, but it should still work. If I was better at Matlab, it would probably be possible there, but for iterative problems, Excel usually works fine.

The engineering I do is relatively simple, but most engineers I know avoid tough math problems like the plague. A finite element analysis is such a powerful and easy to use tool that solving differential equations like this really isn't necessary anymore.

 

[Gerenuk]

Is the heat equation only true if phonon scattering is large? Are there any macroscopic materials for which the average mean path of the phonon's greater then say 1cm? As I understand phonon's scatter when there is material impurities. How pure are the purist bulk lattice structures?

To my understanding the heat equation is a macroscopic phenomenological model, which applies for any heat conducting scattering particles that transport heat with a rate proportional to temperature gradient and area.

In fact I've been told that at room temperature electron heat conduction is much larger than the phonon heat conduction, so phonons don't matter that much directly!
Note that the best heat conducting elements are the best electrical conductors for this reason.

 

[Gerenuk]

Generally, problems like this have the temperature at both ends of the rod constant (or Q constant and you find the resulting temperaures). It helps a lot to clearly define what you are trying to do here. Not knowing the pracitcal purpose of this exercise makes it difficult to see why this approach is relevant.

I'm cooling an aluminum block through a thin rod. One side of the rod has a fixed temperature. The other side of the rod has the aluminum block. I need an estimate for the cooling of the block, if possible without knowledge of its shape.
In my attempt I assumed that on some time scale this block has a constant temperature, but as heat is transferred into it, it's temperature will change.

For the transient case, you would have the length elements horizontally, with time vertically, but it should still work. If I was better at Matlab, it would probably be possible there, but for iterative problems, Excel usually works fine.

I'd need thousands of elements. Also the block is 2D or even 3D so that would be really impractical with Excel. In the end I don't actually want a precise numerical simulation. I need to know if it takes seconds, minutes or hours to cool. I'm just not sure how to include the block without too crude approximations.

 

[GT1]

Thanks for the references. I haven't found these books yet, but I'll make a note. Hope there is a similar problem solved.

Your cooling equation wouldn't work in my case, because the whole point is that the temperature is non uniform. The cooling power isn't constant either, since only the temperature of the cooling device is given.

If knowing the temperature distribution of the body is important to you, solve the 1D conduction equation for the rod to find the temperature on the other end (if you want to ignore convection, if you don't want-solve it as a fin), and then solve the 3D time dependent conduction equation with the end temperature of the rod as one of your boundary conditions for the conduction equation.
The easiest way will be to use software like COMSOL where you just need to sketch the object (or import CAD model of the object), add the boundary conditions and run the numeric solver to get the time dependent temperature distribution of the plate.